Find the center, vertices, and foci of the ellipse with the given equation.
-$$\frac { x ^ { 2 } } { 225 } + \frac { y ^ { 2 } } { 625 } = 1$$
A) Center: $( 0,0 )$; Vertices: $( - 25,0 ) , ( 25,0 )$; Foci: $( - 20,0 ) , ( 20,0 )$
B) Center: $( 0,0 )$; Vertices: $( - 25,0 ) , ( 25,0 )$; Foci: $( 0 , - 15 ) , ( 0,15 )$
C) Center: $( 0,0 )$; Vertices: $( 0 , - 25 ) , ( 0,25 )$; Foci: $( 0 , - 20 ) , ( 0,20 )$
D) Center: $( 0,0 )$; Vertices: $( 0 , - 25 ) , ( 0,25 )$; Foci: $( - 15,0 ) , ( 15,0 )$

Question

Quizplus · Accepted Answer

The given equation is in the standard form of an ellipse, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, with $a^2 = 225$ and $b^2 = 625$. This means $a = 15$ and $b = 25$, indicating the major axis is along the y-axis. The center is at the origin $(0,0)$. The vertices are $(0, \pm b)$ or $(0, \pm 25)$, and the foci are $(0, \pm c)$, where $c = \sqrt{b^2 - a^2} = \sqrt{625 - 225} = \sqrt{400} = 20$. Therefore, the foci are $(0, \pm 20)$.

Find the Center, Vertices, and Foci of the Ellipse with the Given